This invention relates to measurements of particle characteristics using multiply scattered radiation. More particularly, methods for obtaining more precise Frequency Domain Photon Migration (FDPM) data and obtaining from the data particle size distributions, volume fractions, and interparticle force characteristics for interacting particles and polydisperse or multi-modal size distributions are provided.
On-line measurements of particle size distribution and volume fraction of colloidal suspensions are needed in various processes of chemical and other industries. The measurements may be used, for example, to evaluate, optimize, and control manufacturing processes or to control product quality.
On-line particle-sizing methods such as turbidity, angular static scattering and dynamic light scattering, based on light scattering from a single particle, are often used. These methods require considerable dilution of most commercial suspensions to avoid the effects of multiple scattering and particle interaction. Dilution may alter the size distribution of the particles. Otherwise used without dilution, empirical calibration of the instruments with like-suspensions of particles to be sized is necessary. Methods have been proposed to suppress the effects of multiple scattering, such as fiber optic dynamic light scattering (FODLS) and modified static light scattering, but these methods are not well suited for particle sizing of multiple scattering, dense colloidal suspensions.
Recently, new optical particle sizing methods, such as Diffusing Wave Spectroscopy (DWS) and diffuse reflectance and/or transmittance spectroscopy have been used for dense colloidal suspensions. To provide accurate sizing information these methods require accounting for particle interactions at high volume fractions.
In recent years, alternatives to optical sizing methods have been developed. These include acoustic and electroacoustic spectroscopy. Similarly to light scattering methods, the acoustic and electroacoustic theories for dilute systems are relatively complete, but the theories for polydisperse concentrated systems are far from complete. While several models based on first principles have been developed to account for particle interactions in concentrated suspensions, few of the models have been successfully used in practical applications. Also, these acoustic methods require knowledge of the physical properties of each phase and electroacoustic spectroscopy can only be used for charged particles. Better methods for characterizing dense suspensions are needed. Further background information on these methods can be found in the article “Approach for Particle Sizing in Dense Polydisperse Colloidal Suspension Using Multiple Scattered Light,” Langmuir 2001, 17, 6142-6147, which is hereby incorporated by reference herein.
Frequency Domain Photon Migration (FDPM) has been proposed for particle measurements (U.S. Pat. No. 5,818,583, for example). Since this technique measures time-dependent propagation characteristics rather than the amount of light detected, it has the major advantage of not requiring calibration. In addition, FDPM measurements allow determination of absorption and scattering properties independently. Therefore, FDPM allows characterizations of colloidal suspension by scattering measurements that are not biased by changes in light absorption or color of suspending fluids. Because FDPM depends on multiply scattered light, it is restricted to particulate suspensions that are concentrated enough to cause multiple scattering events as radiation is transmitted through the suspension. Such suspensions are typically opaque. Using the FDPM method, both size distribution and volume fraction can be directly obtained by inversion algorithms, where the only physical parameter required a priori is the relative refractive index between the particulate and continuous phases. Since no dilution is necessary to avoid multiple scattering, FDPM is suitable for online monitoring of particle size distribution and volume fraction in many commercial processes that include particles or droplets. FDPM has been successfully used for recovery of particle size distribution and volume fraction in opaque, multiple scattering suspensions of polystyrene and of titanium dioxide, for example (S. M. Richter et al., “Particle Sizing Using Frequency Domain Photon Migration,” Part. Part. Syst. Charact., 15, 9, 1998).
Using visible wavelengths, particles from 50 nm to greater than 1 micron in diameter may be characterized in multiple scattering, non-interacting suspensions. However, at volume fractions larger than about 1-5%, corrections for particle interaction become increasingly necessary. Particle interactions can arise from volume exclusion effects as well as from particle interactions, which arise owing to forces that act between particles. As a results of these interactions, the particle suspension becomes ordered and structured. This order and structure impacts scattering and must not only be used to perform sizing, but when sizing information is available, can be used to determine the nature of particle interactions, (S. M. Richter et al, “Characterization of Concentrated Colloidal Suspensions Using Time Dependent Photon Migration Measurements,” Colloid Surf. A, 172, 163, 2000).
Determination of particle characteristics from FDPM data requires solving mathematically an “ill-posed” inverse problem. Error in the FDPM data used to solve the inversion equations has a large effect on particle size distribution, particle size, and other calculated results. Therefore, there is a need for apparatus and method for determining the FDPM (scattering) data to a high degree of accuracy. Further, a method for indicating the quality of the data is needed, such as with a quality parameter that indicates whether the measurement is precise enough for use in an inverse solution. In application Ser. No. 09/297,895, which is hereby incorporated by reference, measurements at multiple modulation frequencies and two locations were disclosed. Since results are non-linear at multiple frequencies, a non-linear regression method of analysis was necessary. A linear method of analysis and method for treating results of measurements from multiple measurements for maximum accuracy are needed.
In previous work on FDPM, particle size distributions were assumed to be either normal, log-normal or Weibull distributions. A method for determining an unknown and arbitrary particle size distribution is needed for polydisperse and multi-modal suspensions, at non-interacting or interacting (high) volume fractions.
Many suspensions of commercial interest contain volume concentrations of the disperse phase far above the 1-5 percent by volume range, where particles can be considered non-interacting. In addition to the methods for determining volume fraction and particle size distribution when particle interaction must be considered, methods for indicating any attractive or repulsive forces between the interacting particles are needed, since such forces are important in determining stability, rheological properties, and other properties of the high-volume-fraction suspensions. These interparticle forces, along with the volume exclusion effects, cause additional order within the suspension. Therefore, methods for interpreting interparticle forces as result of the order are needed. In applying FDPM to a particular sample containing particles, various models may be used to interpret a measured scattering coefficient in terms of particle concentration, particle size distribution and inter-particle forces. In the method disclosed in Ser. No. 09/297,895, interparticle interactions arising from volume exclusion were interpreted assuming a monodisperse sample. Methods for considering more general particle size distributions are needed. A device that includes a processor to execute computer programs to provide results in terms of the various models is needed.